Mixed Numbers as Quotients (Definition, Types and Examples) – BYJUS

# Mixed Numbers as Quotients

Mixed numbers are numbers that have a whole number part and a fractional part. Learn how they are connected to fractions, how they are expressed, and some methods to represent them diagrammatically....Read MoreRead Less

## What are Mixed Numbers?

Mixed numbers are numbers that have a whole number part and a fractional part. A mixed number represents an improper fraction — a number between two whole numbers. The general form of a fraction is $$\frac{a}{b}$$. In an improper fraction, the numerator is greater than the denominator, that is, a > b. We use mixed fractions in place of improper fractions to easily identify the position of the fraction on the number line. Let’s observe the mixed fraction $$2\frac{3}{4}$$.

In this case, 2 represents the whole number part and $$\frac{3}{4}$$ represents the fractional part. This indicates that the mixed number lies between 2 and 3. Since the denominator of the fraction is 4, each whole is divided into four parts. The numerator of the fraction is 3; we only have three parts of the third whole.

## How Do We Express Mixed Numbers?

Mixed numbers can also be expressed as improper fractions, which are in the form $$\frac{a}{b}$$, where a > b. Consider the fraction $$1\frac{1}{2}$$

$$1\frac{1}{2}$$ can be rewritten as $$\frac{2 {\times}1 + 1}{2}=\frac{2 + 1}{2}=\frac{3}{2}$$.

In this case, $$\frac{3}{2}$$ is an improper fraction. It means the numerator is greater than the denominator (a > b). A fraction whose numerator is greater than the denominator lies outside the region of 0 and 1.

And just like all fractions, a mixed number can also be expressed as $$a\div b$$. So, $$1\frac{1}{2}$$ can also be expressed as $$3\div 2$$.

## Representing Mixed Numbers

Mixed numbers can be expressed as quotients using two types of models: tape diagrams and area models. Let’s take a look at how these models can be used to represent mixed numbers.

Using tape diagram

Consider the following: $$5 \div 2$$.

$$5 \div 2=\frac{5}{2}$$

We know that this number doesn’t lie between 0 and 1, as the numerator is greater than the denominator. We can use a tape diagram to find where this mixed number is located.

We have 5 wholes in this tape diagram. Each of these wholes is divided into two parts. So, each part is  $$\frac{1}{2}$$  of the whole. We get  $$\frac{5}{2}$$  by adding five halves.

Hence, 5 ÷ 2 = $$\frac{5}{2}$$ = 2$$\frac{1}{2}$$

Using area model

$$5 \div 2$$ can also be represented using an area model.

5 wholes are divided into two parts. One-half of the wholes is taken into a single group, and the other half is taken into another group.

## Solved Examples

Example 1:

Use a model to represent $$4\div 3$$.

Solution:

$$4\div 3$$ is the same as $$\frac{4}{3}$$.

We can represent this using a tape diagram by showing 4 wholes and dividing each of these wholes into 3 parts.

Here, 4 wholes are divided into 3 equal parts.

So, each part is $$\frac{1}{3}$$ of the whole.

That means 4 ÷ 3 = $$\frac{4}{3}$$

= $$\frac{3+1}{3}=1\frac{1}{3}$$

Therefore, $$4\div 3=\frac{4}{3}$$ or $$1\frac{1}{3}$$

Example 2:

Use an area model to represent $$5\div 4$$.

Solution:

$$5\div 4$$ can also be written as $$\frac{5} {4}$$. This fraction can be represented using an area model by showing 5 wholes and dividing each of the wholes into 4 parts.

There are 5 wholes and they are divided into four parts. One part from each whole is grouped together.

So, each group gets $$\frac{1} {4}$$ of each whole.

$$5\div 4=\frac{5} {4}=\frac{4+1} {4}$$

= $$1+\frac{1} {4}=1\frac{1} {4}$$

Therefore, $$5\div 4=\frac{5} {4}$$ or $$1\frac{1} {4}$$

Example 3:

You are sharing 8 small pizzas with 4 of your friends. How many whole pizzas will each person get? Also, what fractional amount of a pizza will each person get?

Solution:

As you are sharing these pizzas with 4 of your friends, the pizza is divided among 5 people (including yourself).

To find how much pizza each person gets, we can divide 8 ÷ 5. Let’s express this using an area model.

$$8\div 5=\frac{8} {5}$$

$$\frac{8} {5}=\frac{5+3}{5}=\frac{5}{5}+\frac{3}{5}$$

$$=1+\frac{3}{5}=1\frac{3}{5}$$

Whole number part = 1

Fractional part = $$\frac{3} {5}$$

So, each person gets a whole pizza and $$\frac{3} {5}$$ of another pizza.

Example 4:

If Leo runs 19 miles in 3 hours at a steady pace, is he running more than $$\frac{11}{2}$$ miles in an hour?

Solution:

We know that Leo can run 19 miles in 3 hours. Since he is running at a steady pace, we can say that he runs $$\frac{19}{3}$$ miles per hour.

Leo’s pace = $$\frac{19}{3}$$ miles per hour

Now, we need to compare $$\frac{19}{3}$$  with $$\frac{11}{2}$$ to see if he runs more than $$\frac{11}{2}$$ in an hour.

The easiest way to compare these improper fractions is to convert them into mixed numbers.

$$\frac{19}{3}=\frac{18+1}{3}=\frac{18}{3}+\frac{1}{3}$$

$$=6+\frac{1}{3}=6\frac{1}{3}$$

$$\frac{11}{2}=\frac{10+1}{2}=\frac{10}{2}+\frac{1}{2}$$

$$=5+\frac{1}{2}=5\frac{1}{2}$$

Leo can run $$6\frac{1}{3}$$ miles in an hour. We know that $$6\frac{1}{3}$$ miles is greater than $$5\frac{1}{2}$$ miles.

Hence, Leo runs more than $$5\frac{1}{2}$$ or $$\frac{11}{2}$$ miles in an hour.

The general form of a fraction is $$\frac{a}{b}$$. If b > a or the denominator is greater than the numerator, it is a proper fraction. On the other hand, if a > b or the numerator is greater than the denominator, it is an improper fraction.
A fraction can be converted into a mixed fraction when the numerator is greater than the denominator (a > b in $$\frac{a}{b}$$). In other words, only improper fractions can be converted into mixed fractions.